I want a script which would just paste text from current clipboard and align it in the text frame. But I only figured out how to paste, and align it horizontally. How to do it vertically? So the text fits right in the center of the text frame. Here what I have done so far:
app.paste();
app.selection[0].parentStory.justification = Justification.CENTER_ALIGN;
Vertical justification is a text frame property so you'll need to find the frames corresponding to the story. Here's a snippet that picks up where yours ends.
// Get the parent story of the current selection
var myParentStory = app.selection[0].parentStory;
// Get all text frames for that parent story
var textFrames = myParentStory.textContainers;
// Loop through each text frame to set vertical alignment
for (var i = 0; i < textFrames.length; i++) {
textFrames[i].textFramePreferences.verticalJustification = VerticalJustification.CENTER_ALIGN;
}